What's the difference between arithmetic mean and average? I'm trying to intuitively understand an average / arithmetic mean:
Here's my attempt:

In front of me, I see 1 thermos, two computer mice, two pens, and an iPhone. If I sum those, I get $1+2+2+1=6$ items, $4$ of which are, in functional terms, different. So if I divide $6$ by the number of different items, I get $6/4 = 1.5$ which, I suppose, indicates the amount of repetition. 

Can you explain it better and/or more concretely?
 A: People looking for this question might find the following helpful:

In probability and statistics, mean and expected value are used synonymously to refer to one measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. In the case of a discrete probability distribution of a random variable $X$, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value $x$ of $X$ and its probability $P(x)$, and then adding all these products together, giving $\mu = \sum x P(x)$.

Source: Wikpedia

In colloquial language average usually refers to the sum of a list of numbers divided by the size of the list, in other words the arithmetic mean. However, the word "average" can be used to refer to the median, the mode, or some other central or typical value. In statistics, these are all known as measures of central tendency. Thus the concept of an average can be extended in various ways in mathematics, but in those contexts it is usually referred to as a mean (for example the mean of a function).

Source: Wikpedia

In statistics and probability theory, the median is the numerical value separating the higher half of a data sample, a population, or a probability distribution, from the lower half. 

Source: Wikipedia
A: I am under the impression that you understand how to calculate the arithmetic mean and know what it is technically.  I assume you are asking how to use it, or what it's good for, so let me propose an example.
You're in charge of the IT department for a large company and in 2013 had to handle 8 large projects.  In terms of equipment, consultants and software it cost \$1 million to handle these 8 projects.  The cost of the projects varied, but you know the total, and you know how many projects there were.  The arithmetic mean of the cost of a project is \$1 million divided by 8, which is \$125,000.
Your 2014 project list has 10 projects and you need to budget for them.  All other things being equal you might assume that the average project in 2014 will cost the same as an average project in 2013.  Thus multiplying the 2013 average of \$125,000 by 10 projects, you get a 2014 budget of \$1,250,000.
I'm not sure if this is what you were looking for.  Let me add that, in my opinion, the arithmetic mean is particularly useful for doing calculations.  I think, however, that to get an intuitive understanding of a population (projects, people, gadgets, whatever), the median usually is more helpful.  Also take a look at the mode, another statistic that has an intuitive use.  This is not to say that the median and mode do not also have very good uses in carrying out appropriate calculations.
A: One guy gives you 2 dollars.
Second guy gives you 7 dollars.
Third guy gives you 3 dollars.
One guy gives your friend 4 dollars.
Second guy gives your friend 4 dollars.
Third guy gives your friend 4 dollars.
You both got 12 dollars.  On average, you both got 4 dollars per person.  He also actually got 4 dollars 3 times, so his average is easy to see is 4.  You got different amounts of money, so your average has to be found as 12/3.
